Subgroups — D₁₃⋊#2171 #16807172

Order 3328.

654 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃⋊#504	C₂	1	2	yes	maximal, complemented
2	#13131228	C₂	1	2	yes	maximal, complemented
3	#13131227	C₂	1	4	yes	maximal, complemented
4	#13131226	C₂	1	2	yes	maximal, complemented
5	D₁₃⋊#364	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₃⋊#350	C₂	1	2	yes	maximal, complemented
7	#13131037	C₂	1	1	yes	maximal, characteristic, complemented
8	D₁₃×#443	C₂	1	1	yes	maximal, characteristic
9	D₁₃×#135	C₄	1	1	yes	characteristic
10	D₁₃×#131	C₂²	1	2	yes	—
11	#11719162	C₂²	1	4	yes	complemented
12	#11719096	—	2	4	no	—
13	#11719096	C₂²	1	4	yes	complemented
14	D₁₃⋊M₃₂	—	2	2	no	—
15	D₁₃⋊M₃₂	C₂²	1	2	yes	—
16	C₂×D₁₃⋊C₁₆	—	2	2	no	—
17	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₁₃×#443	C₄	1	1	yes	characteristic
20	#11718857	C₂²	1	1	yes	characteristic
21	#11718805	C₄	1	2	yes	—
22	#11718556	C₂²	1	1	yes	characteristic
23	D₁₃×#108	C₄	1	1	yes	characteristic
24	#11718485	C₂²	1	2	yes	—
25	D₁₃×#104	C₄	1	2	yes	—
26	#11718361	C₄	1	1	yes	characteristic
27	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
28	#11718304	C₂²	1	4	yes	complemented
29	#11718270	C₂²	1	4	yes	complemented
30	#11718261	C₂²	1	2	yes	complemented
31	#11718258	C₂²	1	2	yes	complemented
32	#11718233	C₂²	1	2	yes	complemented
33	#11718231	C₂²	1	2	yes	complemented
34	#41×D₁₃	C₂×C₄	1	2	yes	—
35	C₁₃×#135	C₂×C₄	1	1	yes	characteristic
36	C₁₃×#131	C₂×C₄	1	2	yes	—
37	#88204	C₂×C₄	1	2	yes	—
38	#88202	C₂³	1	2	yes	—
39	#88173	C₈	1	1	yes	characteristic
40	Q₁₆×D₁₃	C₈	1	1	yes	characteristic
41	#88170	C₈	1	2	yes	—
42	SD₁₆×D₁₃	C₈	1	2	yes	—
43	#88166	C₈	1	1	yes	characteristic
44	D₈×D₁₃	C₈	1	1	yes	characteristic
45	#88161	—	2	2	no	—
46	#88161	C₂³	1	2	yes	—
47	M₁₆×D₁₃	—	2	2	no	—
48	M₁₆×D₁₃	C₂×C₄	1	2	yes	—
49	#88158	C₂×C₄	1	1	yes	characteristic
50	#88155	—	2	2	no	—
51	#88155	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₁₃	—	2	2	no	—
53	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
55	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
56	#88106	—	2	4	no	—
57	#88106	C₂³	1	4	yes	—
58	C₂×#6092	—	2	4	no	—
59	C₂×#6092	C₂³	1	2	yes	—
60	C₂×#6092	C₂³	1	2	yes	—
61	#88098	—	2	2	no	—
62	#88098	—	2	4	no	—
63	D₁₃⋊C₁₆	—	2	2	no	—
64	D₁₃⋊C₁₆	—	2	1	no	—
65	D₁₃⋊C₁₆	—	2	1	no	—
66	D₁₃⋊C₁₆	—	2	4	no	—
67	D₁₃⋊C₁₆	D₄	1	2	yes	—
68	D₁₃⋊C₁₆	D₄	1	2	yes	—
69	C₁₃×#108	C₂×C₄	1	1	yes	characteristic
70	C₁₃×#104	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
72	#88077	C₂×C₄	1	2	yes	—
73	#88065	C₂×C₄	1	2	yes	—
74	#88062	C₂×C₄	1	2	yes	—
75	#88059	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
78	#88045	C₂×C₄	1	2	yes	—
79	C₄×#1055	C₂³	1	1	yes	characteristic
80	C₂×#2171	—	13	1	no	maximal, Sylow
81	C₁₃×#41	C₂²×C₄	1	2	yes	—
82	#6131	C₂×C₈	1	2	yes	—
83	Q₈×D₁₃	C₂×C₈	1	2	yes	—
84	#6129	—	2	2	no	—
85	#6129	C₂×C₈	1	2	yes	—
86	D₄×D₁₃	—	2	2	no	—
87	D₄×D₁₃	C₂×C₈	1	2	yes	—
88	#6127	—	2	2	no	—
89	#6127	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₁₃	—	2	2	no	—
91	C₂×C₄×D₁₃	—	2	1	no	—
92	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₁₃×Q₁₆	C₂×C₈	1	1	yes	characteristic
94	C₁₃×SD₁₆	C₂×C₈	1	2	yes	—
95	C₁₃×D₈	C₂×C₈	1	1	yes	characteristic
96	C₁₃×M₁₆	—	2	2	no	—
97	C₁₃×M₁₆	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	#6107	C₂×C₈	1	2	yes	—
103	#6106	C₂×C₈	1	2	yes	—
104	#6105	C₂×C₈	1	2	yes	—
105	#6104	C₂×C₈	1	2	yes	—
106	C₄×Dic₁₃	—	2	1	no	—
107	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	#6099	—	2	2	no	—
109	#6099	C₂²×C₄	1	2	yes	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
112	C₂×#1055	C₂⁴	1	1	yes	characteristic
113	Dic₅₂	C₂×C₈	1	1	yes	characteristic
114	D₁₀₄	C₂×C₈	1	1	yes	characteristic
115	#6095	C₂×C₈	1	2	yes	—
116	#6094	—	2	2	no	—
117	#6094	—	2	2	no	—
118	#6094	—	2	2	no	—
119	C₈×D₁₃	—	2	2	no	—
120	C₈×D₁₃	—	2	2	no	—
121	C₈×D₁₃	—	2	2	no	—
122	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
123	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
124	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
125	C₈×D₁₃	C₂×C₈	1	1	yes	characteristic
126	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
127	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
128	C₈×D₁₃	#41	1	1	yes	characteristic
129	C₈×D₁₃	#41	1	1	yes	characteristic
130	#6092	—	2	4	no	—
131	#6092	—	2	2	no	—
132	#6092	C₂×D₄	1	4	yes	—
133	C₂×#504	—	13	2	no	—
134	C₂×#443	—	13	1	no	—
135	#2171	—	13	2	no	—
136	#2171	—	13	4	no	—
137	#2171	—	13	2	no	—
138	C₂×#364	—	13	1	no	—
139	C₂×#350	—	13	2	no	—
140	C₂×#346	—	13	1	no	—
141	C₂²×D₁₃	—	2	2	no	—
142	C₂²×D₁₃	#131	1	1	yes	characteristic
143	Q₈×C₁₃	C₂²×C₈	1	2	yes	—
144	D₄×C₁₃	—	2	2	no	—
145	D₄×C₁₃	C₂²×C₈	1	2	yes	—
146	C₂×C₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	1	no	—
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂²⋊D₁₃	—	2	2	no	—
150	C₂²⋊D₁₃	—	2	2	no	—
151	C₂²⋊D₁₃	—	2	2	no	—
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	—	2	1	no	—
154	C₂×Dic₁₃	#131	1	1	yes	characteristic
155	D₅₂	—	2	2	no	—
156	D₅₂	C₂²×C₈	1	2	yes	—
157	C₄×D₁₃	—	2	2	no	—
158	C₄×D₁₃	—	2	2	no	—
159	C₄×D₁₃	—	2	2	no	—
160	C₄×D₁₃	—	4	2	no	—
161	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
162	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
163	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
164	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
165	Dic₂₆	—	2	2	no	—
166	Dic₂₆	C₂²×C₈	1	2	yes	—
167	C₈×C₁₃	—	2	2	no	—
168	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
169	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
170	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
171	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
172	#1055	—	2	2	no	—
173	#1055	C₂²×C₈	1	1	yes	characteristic
174	#1055	C₂²×C₈	1	1	yes	characteristic
175	#1055	C₂²×D₄	1	1	yes	characteristic
176	#1055	C₂×#41	1	1	yes	characteristic
177	C₂×#135	—	13	1	no	—
178	C₂×#131	—	13	2	no	—
179	#504	—	13	4	no	—
180	#504	—	13	4	no	—
181	#504	—	26	4	no	—
182	C₂×M₃₂	—	13	2	no	—
183	C₂×M₃₂	—	26	2	no	—
184	C₂²×C₁₆	—	13	1	no	—
185	C₂²×C₁₆	—	13	1	no	—
186	C₂²×C₁₆	—	26	2	no	—
187	#443	—	13	1	no	—
188	#443	—	13	2	no	—
189	#443	—	13	1	no	—
190	#443	—	13	2	no	—
191	#443	—	13	1	no	—
192	#443	#261	1	1	yes	characteristic
193	C₂×#108	—	13	1	no	—
194	C₂×#104	—	13	2	no	—
195	C₂×C₄×C₈	—	13	1	no	—
196	#364	—	13	2	no	—
197	#364	—	13	2	no	—
198	#350	—	13	4	no	—
199	#350	—	13	4	no	—
200	#346	—	13	2	no	—
201	#346	—	13	2	no	—
202	C₂²×C₁₃	—	2	2	no	—
203	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
204	C₂×D₁₃	—	2	2	no	—
205	C₂×D₁₃	—	2	2	no	—
206	C₂×D₁₃	—	4	4	no	—
207	C₂×D₁₃	C₈×D₄	1	1	yes	characteristic
208	C₂×D₁₃	C₈×D₄	1	1	yes	characteristic
209	C₄×C₁₃	—	2	2	no	—
210	C₄×C₁₃	—	4	1	no	—
211	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
212	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
213	Dic₁₃	—	2	2	no	—
214	Dic₁₃	—	4	1	no	—
215	Dic₁₃	C₈×D₄	1	1	yes	characteristic
216	Dic₁₃	C₈×D₄	1	1	yes	characteristic
217	C₂×#41	—	13	2	no	—
218	#135	—	13	2	no	—
219	#135	—	13	1	no	—
220	#135	—	13	2	no	—
221	#135	—	13	1	no	—
222	#135	—	13	1	no	—
223	#135	C₂×#261	1	1	yes	characteristic
224	C₂×Q₁₆	—	13	1	no	—
225	C₂×SD₁₆	—	13	2	no	—
226	C₂×D₈	—	13	1	no	—
227	#131	—	13	2	no	—
228	#131	—	13	2	no	—
229	#131	—	13	2	no	—
230	#131	—	26	2	no	—
231	#131	—	26	2	no	—
232	#131	C₂×#261	1	2	yes	—
233	C₂×M₁₆	—	13	2	no	—
234	C₂×M₁₆	—	26	2	no	—
235	C₂²×C₈	—	13	1	no	—
236	C₂²×C₈	—	13	1	no	—
237	C₂²×C₈	—	26	2	no	—
238	C₂×C₄²	—	13	1	no	—
239	M₃₂	—	13	4	no	—
240	M₃₂	—	26	4	no	—
241	M₃₂	—	26	4	no	—
242	M₃₂	—	26	2	no	—
243	C₂×C₁₆	—	13	2	no	—
244	C₂×C₁₆	—	13	2	no	—
245	C₂×C₁₆	—	13	2	no	—
246	C₂×C₁₆	—	13	2	no	—
247	C₂×C₁₆	—	26	4	no	—
248	C₂×C₁₆	—	26	4	no	—
249	C₂×C₁₆	—	26	1	no	—
250	C₂×C₁₆	—	26	1	no	—
251	C₂×C₁₆	—	26	2	no	—
252	#108	—	13	2	no	—
253	#108	—	13	1	no	—
254	#108	C₂×#261	1	1	yes	characteristic
255	#104	—	13	2	no	—
256	#104	—	13	2	no	—
257	#104	—	13	2	no	—
258	#104	C₂×#261	1	2	yes	—
259	C₄×C₈	—	13	1	no	—
260	C₄×C₈	—	13	1	no	—
261	C₄×C₈	—	13	1	no	—
262	C₄×C₈	C₂×#261	1	1	yes	characteristic
263	C₂×C₁₃	—	2	1	no	—
264	C₂×C₁₃	—	4	2	no	—
265	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
266	D₁₃	—	2	1	no	—
267	D₁₃	—	4	2	no	—
268	D₁₃	#2171	1	2	yes	complemented
269	#41	—	13	2	no	—
270	#41	—	13	2	no	—
271	#41	—	13	2	no	—
272	#41	—	26	2	no	—
273	#41	—	26	2	no	—
274	#41	C₂²×#261	1	2	yes	—
275	C₂×Q₈	—	13	2	no	—
276	C₂×D₄	—	13	2	no	—
277	C₂×D₄	—	26	2	no	—
278	C₂²×C₄	—	13	1	no	—
279	C₂²×C₄	—	26	2	no	—
280	C₂²×C₄	—	26	1	no	—
281	Q₁₆	—	13	2	no	—
282	Q₁₆	—	13	1	no	—
283	Q₁₆	D₁₃⋊C₈	1	1	yes	characteristic
284	SD₁₆	—	13	2	no	—
285	SD₁₆	—	13	2	no	—
286	SD₁₆	—	13	2	no	—
287	SD₁₆	D₁₃⋊C₈	1	2	yes	—
288	D₈	—	13	2	no	—
289	D₈	—	13	1	no	—
290	D₈	D₁₃⋊C₈	1	1	yes	characteristic
291	M₁₆	—	2	2	no	—
292	M₁₆	—	13	2	no	—
293	M₁₆	—	26	2	no	—
294	M₁₆	—	26	2	no	—
295	M₁₆	—	26	2	no	—
296	M₁₆	—	26	2	no	—
297	M₁₆	C₂²×#261	1	2	yes	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	13	1	no	—
300	C₂×C₈	—	13	1	no	—
301	C₂×C₈	—	13	1	no	—
302	C₂×C₈	—	13	1	no	—
303	C₂×C₈	—	13	1	no	—
304	C₂×C₈	—	13	1	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	13	1	no	—
308	C₂×C₈	—	13	1	no	—
309	C₂×C₈	—	26	2	no	—
310	C₂×C₈	—	26	2	no	—
311	C₂×C₈	—	26	2	no	—
312	C₂×C₈	—	26	2	no	—
313	C₂×C₈	C₂²×#261	1	1	yes	characteristic
314	C₂×C₈	C₂²×#261	1	1	yes	characteristic
315	C₄²	—	13	1	no	—
316	C₄²	—	26	1	no	—
317	C₄²	C₂²×#261	1	1	yes	characteristic
318	C₁₆	—	13	4	no	—
319	C₁₆	—	26	2	no	—
320	C₁₆	—	26	4	no	—
321	C₁₃	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
322	C₂³	—	13	1	no	—
323	C₂³	—	26	2	no	—
324	Q₈	—	13	2	no	—
325	Q₈	—	26	2	no	—
326	Q₈	C₂×D₁₃⋊C₈	1	2	yes	—
327	D₄	—	2	2	no	—
328	D₄	—	13	2	no	—
329	D₄	—	26	2	no	—
330	D₄	—	26	2	no	—
331	D₄	—	26	2	no	—
332	D₄	—	26	2	no	—
333	D₄	C₂×D₁₃⋊C₈	1	2	yes	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	13	1	no	—
337	C₂×C₄	—	13	1	no	—
338	C₂×C₄	—	13	1	no	—
339	C₂×C₄	—	13	1	no	—
340	C₂×C₄	—	13	1	no	—
341	C₂×C₄	—	26	2	no	—
342	C₂×C₄	—	26	2	no	—
343	C₂×C₄	—	26	2	no	—
344	C₂×C₄	—	26	2	no	—
345	C₂×C₄	—	26	1	no	—
346	C₂×C₄	—	52	2	no	—
347	C₂×C₄	C₂³×#261	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	13	1	no	—
350	C₈	—	13	1	no	—
351	C₈	—	13	1	no	—
352	C₈	—	13	1	no	—
353	C₈	—	26	2	no	—
354	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
355	C₈	C₂×D₁₃⋊C₈	1	1	yes	characteristic
356	C₈	D₄×#261	1	1	yes	characteristic
357	C₈	D₄×#261	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	13	1	no	—
360	C₂²	—	13	1	no	—
361	C₂²	—	26	2	no	—
362	C₂²	—	26	2	no	—
363	C₂²	—	52	4	no	—
364	C₂²	D₁₃⋊#131	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	4	1	no	—
367	C₄	—	13	1	no	—
368	C₄	—	13	1	no	—
369	C₄	—	26	2	no	—
370	C₄	—	52	1	no	—
371	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
372	C₄	C₂×D₄×#261	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	4	2	no	—
375	C₂	—	13	2	no	—
376	C₂	—	26	1	no	—
377	C₂	—	52	2	no	—
378	C₂	D₄×D₁₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#443	C₄
socle	C₂×C₁₃	C₂×C₈×D₄
prime radical	D₁₃×#135	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#443	#261
hypocenter	C₁₃	C₂×#2171
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. D₁₃⋊#504	C₂	semidirect
2. #13131228	C₂	semidirect
3. #13131227	C₂	semidirect
4. #13131226	C₂	semidirect
5. D₁₃⋊#364	C₂	semidirect
6. D₁₃⋊#350	C₂	semidirect
7. #13131037	C₂	semidirect
11. #11719162	C₂²	semidirect
13. #11719096	C₂²	semidirect
28. #11718304	C₂²	semidirect
29. #11718270	C₂²	semidirect
30. #11718261	C₂²	semidirect
31. #11718258	C₂²	semidirect
32. #11718233	C₂²	semidirect
33. #11718231	C₂²	semidirect
268. D₁₃	#2171	semidirect
321. C₁₃	C₂×#2171	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₁₃⋊#504	2
#13131228	2
#13131227	2
#13131226	2
D₁₃⋊#364	2
D₁₃⋊#350	2
#13131037	2
D₁₃×#443	2
C₂×#2171	13

Subgroups — D13⋊#2171 #16807172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃⋊#2171 #16807172